The properties of the $S$-wave $D_s\bar{D}_s$ bound state

TL;DR

This paper addresses whether a near-threshold $D_s\bar{D}_s$ bound state exists and how it decays. It employs the Bethe–Salpeter equation in the ladder and instantaneous approximations, with a kernel from vector-meson ($\phi$ and $J/\psi$) exchanges and a monopole form factor to handle off-shell effects, solving for the S-wave bound-state wavefunction $\chi_P(\mathbf{p})$. The main finding is that a bound state arises only when both exchanges are included, with the binding characterized by a small energy $E_b$ and a coupling sensitivity to the cutoff parameter $\alpha$. Using the normalized BS wavefunctions, the authors compute partial decay widths to $D\bar{D}$, $\eta_c\eta$, and $J/\psi\omega$, finding that $D\bar{D}$ dominates due to lighter exchange propagators, while the other channels are strongly suppressed; the total widths increase with binding energy, consistent with a loosely bound molecular interpretation relevant to the $X(3915)/X(3960)$ phenomenology. These results provide concrete, testable predictions for experimental searches of a $D_s\bar{D}_s$ molecule and contribute to the understanding of near-threshold hadronic bound states in the charm sector.

Abstract

In this work, we investigate possible bound states of the $D_s\bar{D}_s$ system in the Bethe-Salpeter formalism in the ladder and instantaneous approximations. By numerically solving the Bethe-Salpeter equation with a kernel that includes the contributions from $φ$ and $J/ψ$ exchanges, we confirm the existence of a bound state in the $D_s\bar{D}_s$ system. We further investigate the partial decay widths of the $D_s\bar{D}_s$ bound state into $D\bar{D}$, $η_cη$, and $J/ψω$, finding that these partial widths are sensitive to the parameter $α$ in our model. Notably, we observe that the dominant decay channel for the $D_s\bar{D}_s$ bound state is that into $D\bar{D}$.

Figures (4)

• Figure 1: The possible bound states in the $D_s\bar{D}_s$ system.
• Figure 2: The particle decay width of the $D_s\bar{D}_s$ bound state to $D\bar{D}$ is shown for $E_b$ = 1 MeV (a), $E_b$ = 5 MeV (b), and $E_b$ = 15 MeV (c) across the allowed parameter $\alpha$.
• Figure 3: The particle decay width of the $D_s\bar{D}_s$ bound state to $\eta_c\eta$ is shown for $E_b$ = 1 MeV (a), $E_b$ = 5 MeV (b), and $E_b$ = 15 MeV (c) across the allowed parameter range of $\alpha$.
• Figure 4: The decay width of the $D_s\bar{D}_s$ bound state to $J/\psi\omega$ is shown for $E_b$ = 1 MeV (a), $E_b$ = 5 MeV (b), and $E_b$ = 15 MeV (c), across the allowed range of the parameter $\alpha$.